Reciprocal calibration for channel estimation based on second-order statistics

ABSTRACT

A wireless communication method includes receiving, by a first wireless device during a training phase, reference tones using a first number of resource elements from a transmitter of a second wireless device, wherein the first wireless device comprises multiple receiving antennas, estimating, by the first wireless device, from the receiving the reference tones, a second order statistics of wireless channels between the multiple receiving antennas and the transmitter of the second wireless device, and performing channel estimation, during an operational phase subsequent to the training phase, using the second order statistics and reference tones received on a second number of resource elements, wherein the second number is less than the first number.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This patent document is a continuation of U.S. patent application Ser.No. 17/251,765, filed Dec. 11, 2020 which is a 371 of InternationalApplication No. PCT/US2019/037095, filed Jun. 13, 2019, which claimspriority to and benefits of U.S. Provisional Application No. 62/684,594entitled “SECOND ORDER STATISTICS FOR EFFICIENT CHANNEL ESTIMATION,”filed on Jun. 13, 2018 and U.S. Provisional Application No. 62/726,822entitled “RECIPROCITY CALIBRATION IN WIRELESS COMMUNICATIONS,” filed onSep. 4, 2018. The entire contents of the aforementioned patentapplications are incorporated by reference as part of the disclosure ofthis patent document.

TECHNICAL FIELD

The present document relates to wireless communication, and moreparticularly, precoding of wireless signals for transmission.

BACKGROUND

Due to an explosive growth in the number of wireless user devices andthe amount of wireless data that these devices can generate or consume,current wireless communication networks are fast running out ofbandwidth to accommodate such a high growth in data traffic and providehigh quality of service to users.

Various efforts are underway in the telecommunication industry to comeup with next generation of wireless technologies that can keep up withthe demand on performance of wireless devices and networks. Many ofthose activities involve situations in which a large number of userdevices may be served by a network.

SUMMARY

This document discloses techniques for channel estimation. In oneexample, a reciprocal calibration technique is used. In another example,a minimal set of resource elements and an estimate of second orderstatistics of the channel are used.

In an example aspect, a wireless communication method is disclosed. Themethod includes receiving, by a first wireless device during a trainingphase, reference tones using a first number of resource elements from atransmitter of a second wireless device, wherein the first wirelessdevice comprises multiple receiving antennas; estimating, by the firstwireless device, from the receiving the reference tones, a second orderstatistics of wireless channels between the multiple receiving antennasand the transmitter of the second wireless device; and performingchannel estimation, during an operational phase subsequent to thetraining phase, using the second order statistics and reference tonesreceived on a second number of resource elements, wherein the secondnumber is less than the first number.

In another example aspect, a method of wireless communication isdisclosed. The method includes transmitting, to multiple receiveantennas of a first wireless device from a transmit antenna of a secondwireless device, during a training phase, reference tones using a firstnumber of resource elements of a wireless channel between the transmitantenna and the multiple receive antennas; receiving, at an end of thetraining phase, an instruction to transmit reference tones using asecond number of resource elements; and transmitting, during anoperational phase after the training phase, reference tones to themultiple receive antennas of the first wireless device using the secondnumber of resource elements, wherein the second number is different fromthe first number and wherein the second number is based on an estimatedsecond order statistics of the wireless channel.

In yet another example aspect, a method of wireless communication isdisclosed. The method includes estimating, during a training phase, asecond order statistics for a first wireless channel and a secondwireless channel between a transmitter and a receiver comprisingmultiple antennas, wherein the second order statistics is estimatedusing reference tones transmitted on a first number of resourceelements; predicting, during an operational phase subsequent to thetraining phase, an estimate of the second wireless channel based on thesecond order statistics and an estimate of the first wireless channelcalculated using reference tones transmitted on a second number ofresource elements, where the second number is less than the firstnumber; and communicating, during the operational phase, over the secondwireless channel using the estimate of the second wireless channelresulting from the predicting; wherein the first wireless channel andthe second wireless channel include non-overlapping frequencies.

In yet another example aspect, a method of wireless communication isdisclosed. The method includes receiving, by a first communicationdevice, a number of subcarriers from a second communication device, eachincluding a corresponding reference signal; calculating an inversionfactor for each subcarrier based on a received value of thecorresponding reference signal; and transmitting by the firstcommunication device to the second communication device, at least someof the subcarriers by scaling a pilot signal using a correspondinginversion factor.

In yet another example aspect, a method of wireless communication isdisclosed. The method includes transmitting, to a first communicationdevice, from a second communication device, a number of subcarriers,each subcarrier including a corresponding reference signal; receiving,from the first communication device, at least some of the subcarrierscarrying pilot signals scaled by inversions factors for the subcarriers;and estimating a communication channel between the second communicationdevice to the first communication device using the inversion factors.

In yet another example aspect, a wireless communication apparatus thatimplements the above-described methods is disclosed.

In yet another example aspect, the methods may be embodied asprocessor-executable code and may be stored on a computer-readableprogram medium.

In yet another example aspect, a wireless communication system thatoperates by providing a single pilot tone for channel estimation isdisclosed.

These, and other, features are described in this document.

DESCRIPTION OF THE DRAWINGS

Drawings described herein are used to provide a further understandingand constitute a part of this application. Example embodiments andillustrations thereof are used to explain the technology rather thanlimiting its scope.

FIG. 1 shows an example of a fixed wireless access network.

FIG. 2 shows another example of a fixed wireless access network.

FIG. 3A shows an example of a wireless channel between a transmitter anda receiver device.

FIG. 3B shows another example of a wireless channel between atransmitter and a receiver device.

FIG. 3C shows an example of a wireless channel between a single transmitantenna and multiple receive antennas.

FIG. 4 is a graph showing an example of effect of number of tones usedfor reference signal transmissions on average error covariance.

FIG. 5 shows an example of a mobile wireless communication network witha multi-antenna base-station and multiple user antennas.

FIG. 6 shows an example of FDD prediction setup, wherein thebase-station uses the channel response of the uplink to predict thechannel response of the downlink in a different frequency and on a latersubframe.

FIG. 7 shows examples of bandwidth partitions.

FIG. 8 shows an example of a bandwidth partition with the same timeinterval for reference signal (RS) 1 and RS2.

FIG. 9 shows an example of a bandwidth partition with different timeintervals for reference signal (RS) 1 and RS2.

FIG. 10 shows an example of channel prediction over the same timeinterval.

FIG. 11 shows an example of channel prediction over a different timeinterval.

FIG. 12 shows an example of a wireless channel with reciprocity.

FIG. 13 shows an example antenna configuration in which four transmitand four receive antennas are used at a network-side apparatus.

FIG. 14 shows an example antenna configuration in a user-sidecommunications apparatus.

FIG. 15 shows a block diagram for an example implementation ofreciprocity calculation.

FIG. 16 shows an example of a wireless transceiver apparatus.

FIG. 17 shows an example communication network.

FIG. 18 is a flowchart for an example wireless communication method.

FIG. 19 is a flowchart for another example wireless communicationmethod.

FIG. 20 is a flowchart for another example wireless communicationmethod.

FIG. 21 is a flowchart for another example wireless communicationmethod.

FIG. 22 is a flowchart for another example wireless communicationmethod.

DETAILED DESCRIPTION

To make the purposes, technical solutions and advantages of thisdisclosure more apparent, various embodiments are described in detailbelow with reference to the drawings. Unless otherwise noted,embodiments and features in embodiments of the present document may becombined with each other.

Section headings are used in the present document to improve readabilityof the description and do not in any way limit the discussion orembodiments (and/or implementations) to the respective sections only.

Channel knowledge is an important component in wireless communication,whether it is for a receiver to equalize and decode the received signal,or for a multi-antenna transmitter to generate a more efficient precodedtransmission.

Typically, channel knowledge is acquired by transmitting known referencesignals (pilots) and interpolating observations and results obtained byreceiving them at the receiver over the entire bandwidth and time ofinterest. The density of the reference signals may depend on thecharacteristics of the channel. Higher delay spreads and higher Dopplerspreads may require denser reference signals along frequency and/or timerespectively, thus, occupying a considerable amount of the availablecapacity.

For example, consider a downlink multi-carrier transmission from abase-station to multiple users. In this case, depending on channelconditions, the reference signals should be dense enough in thetime-frequency grid to allow the users to correctly estimate the channeland equalize the data (e.g., the estimation error due to interpolationof channel estimates along the time-frequency grid should be minimized).Another example is that of an uplink transmission from multi-users to abase station equipped with multi-antennas, where each user needs toallocate resource elements for its own reference signals.

In this document, we propose (i) an efficient method for estimating thechannel from a very small set of resource elements, using thesecond-order statistics of the channel, and (ii) using the second-orderstatistics to reciprocally calibrate the wireless channel. Theeffectiveness of (i) increases with increased number of antennas at thebase-station. As further described below, in some cases, it may bepossible to send a single reference signal (tone) from the transmitantenna to each receive antenna at the base station, and be able todetermine channel at different times and in different frequency bandsbased on the channel estimate obtained from the single (or, in general,reduced number of) reference tones. The efficacy of (ii) is based onderiving knowledge of the non-reciprocal parts of a channel between twonodes, and adjusting the channel response such that precoding may beused for the transmissions.

In the description, the example of a fixed wireless access (FWA) systemis used only for illustrative purpose and the disclosed techniques canapply to other wireless networks, such as cellular or mobilecommunication networks further described in the present document.

FIG. 1 shows an example of a fixed wireless access system 100. A hub102, that includes a transmission facility such as a cell tower, isconfigured to send and receive transmissions to/from multiple locations104. For example, the locations could be user premises or businessbuildings. As described throughout this document, the disclosedembodiments can achieve very high cell capacity fixed wireless access,when compared to traditional fixed access technology. Some techniquesdisclosed herein can be embodied in implementations at the hub 102 or attransceiver apparatus located at the locations 104.

FIG. 2 shows yet another configuration of a fixed access wirelesscommunication system 200 in which hops are used to reach users. Forexample, one cell tower may transmit/receive from another cell tower,which would then relay the transmissions between the principle celltower and the users, thus extending range of the fixed wireless accesssystem. A backhaul may connect the transmission tower 202 with anaggregation router. For example, in one configuration, a 10 Gbps fiberconnection may be used to feed data between a base station at a hub anda fiber hub aggregation router. In one advantageous aspect, deploymentof this technology can be achieved without having to change any networkbandwidth characteristics for harder to reach areas by using thehub/home access point (AP) configuration as a launch point. Sometechniques disclosed herein can be embodied in implementations at themacro tower 202 or at transceiver apparatus located at the otherlocations.

1. Introduction to Embodiments for Reciprocal Calibration for ChannelEstimation Based on Second-Order Statistics

FIGS. 3A, 3B and 3C show simplified wireless networks to highlightcertain aspects of the disclosed technology. A transmitter transmitswireless signals to a receiver in the wireless network. Sometransmissions in the network, variously called as downstream or downlinktransmissions, a network-side node such as a base station acts as atransmitter of wireless signals and one or more user devices act as thereceiver of these wireless signals. For some other transmissions, asdepicted in FIG. 3A, the direction of transmission may reverse. Suchtransmissions are often called uplink or upstream transmissions. Forsuch transmissions, one or more user devices act as transmitters of thewireless signals and a network-side node such as the base station actsas the receiver of these signals (as depicted in FIG. 3A). Other type oftransmissions in the network may include device-to-device transmissions,sometimes called direct or sideband transmissions. In frequency divisionmultiplexing (FDM) networks (also called frequency division duplexing orFDD networks), the transmissions to a base station and the transmissionsfrom the base station may occupy different frequency bands (each ofwhich may occupy continuous or discontinuous spectrum). In time divisionmultiplexing (TDM) networks (also called time division duplexing, orTDD, networks), the transmissions to a base station and thetransmissions from the base station occupy a same frequency band, butare separated in time domain using a TDM mechanism such as time slotbased transmissions. Other types of multiplexing are also possible(e.g., code division multiplexing, orthogonal time frequency space, orOTFS, multiplexing, spatial multiplexing, etc.). In general, the variousmultiplexing schemes can be combined with each other.

From the early days of single antenna systems in which the transmittinguser device had a single transmission antenna and the receiving basestation had a single receiving antenna for that signal, wireless systemshave now evolved to configurations in which transmitters have multipletransmit antennas and receivers have multiple receive antennas. Suchsystems are often called multiple input multiple output (MIMO) systems.In a MIMO system, a wireless channel may therefore be representable as amatrix that acts on a vector representing the multiple inputs andproduces the vector representing the multiple outputs. Depending onchannel conditions, some entries of the channel matrix may change overtime.

The wireless systems described in this document, subjected to someinterference, are known to be able to achieve the channel capacity ifthe interference is known to the transmitter. By using a technique knownas dirty paper coding (DPC), the transmitter can precode a transmissionsuch that the signal arriving at the receiver is interference-free. Inreality, a more implementation-friendly approximation of DPC known asTomlinson-Harashima Precoding (THP) is likely to be used (and isdiscussed in a subsequent section of this document).

A precoded transmission is based on the knowledge of the exact channelresponse between the transmitting antenna(s) of a first terminal denotedby A—typically a base-station (BS)—to the receiving antenna(s) of asecond terminal denoted by B—typically a piece of Consumer PremisesEquipment (CPE) or a user equipment (UE). This channel response can beconsidered to be composed of three different parts as illustrated inFIG. 3B. First, the channel response of the transmitter in terminal A.Second, the channel response of the different reflectors. Third, thechannel response of the receiver in terminal B. The transmitter channelresponses may be due to the transmit chain circuitry such as poweramplifiers, digital to analog converters (DACs), frequency upconvertersand so on. The receiver channel response may be due to receiver-sidecircuitry such as low noise block (LNB), frequency downconverter, analogto digital conversion circuitry (ADC).

1.A. Second-Order Statistics

Without any loss of generality, let's consider a basic setup with atransmitter (e.g., a user device) using a single antenna and a receiver(e.g., a base station) with L antennas. If the transmitter (e.g., userdevice) has more than one antenna, then each antenna can be referred toas a separate transmitter.

Prior to the transmission of data, the system performs a preliminarytraining phase, where the second-order statistics of the channelsbetween the transmitter and each one of the L receiver antennas arecomputed.

FIG. 3C shows an example of the basic setup in which the receivercomprises L receive antennas and the transmitter comprises a singletransmit antenna. The signal transmitted from the transmitter isreflected (and dispersed) by a number of reflectors that give thewireless channel its characteristics. As further discussed in thisdocument, the reflectors may be static or mobile, and have time-variantreflective characteristics.

In this initial phase, or training phase, the transmitter sends knownreference signals, enabling the receiver to estimate the channel acrossall the bandwidth of interest and all the antennas. One example is in amulti-carrier scenario where the bandwidth of interest is representedwith N_(f) discrete tones, at time instance k the receiver will estimatethe channel response h_(k) as a complex column vector of size N=N_(f)·L.This step may be repeated N_(SOS) times and afterwards, the second-orderstatistics may be computed. Two different methods for this computationare proposed here as an example:

Direct method—In this method, the covariance matrix is calculated byarranging all channel responses as columns of the transfer matrix H andcomputing the cross correlation as:

H=[h ₁ |h ₂ | . . . |h _(N) _(SOS) ]  (1)

R _(HH) =H·H*  (2)

Average method—using this method, an average over N_(SOS) of thevariance of each channel response are calculated.

$\begin{matrix}{R_{HH} = {\frac{1}{N_{SOS}}{\sum_{k = 1}^{N_{SOS}}{h_{k} \cdot h_{k}^{*}}}}} & (3)\end{matrix}$

The covariance matrix R_(HH) is large with dimensions N×N. For example,in a 4 antenna system in which 8 tones are used for reference signaltransmissions, N=32, and thus R_(HH) is a 32×32 matrix. However, thematrix encodes only a very small amount of information originating fromthe delay-Doppler profile and angle of arrival/departure of thereflectors of the wireless channel. These essential parameters of thechannel can be extracted from the covariance matrix using a mathematicaloperation such as the Principal Component Analysis (PCA). For example,these parameters may be identified by computing the dominant eigenvaluesand their corresponding eigenvectors of R_(HH). For extracting Kdominant reflectors, R_(HH) may be approximated by

R _(HH) ≈V·D·V*  (4)

where V is an N×K eigenvectors matrix and D is the corresponding K×Keignenvalues diagonal matrix.

1.B. Channel Estimation Using a Small Set of Resource Elements

Once the second-order statistics of the channels between the transmitterand the receivers' L antennas are computed, the transmitter (e.g., asingle transmit antenna of a user device) may transmit only a very smallset of known reference signals, from which the receiver will be able toestimate all the channels (to all the antennas and across all thebandwidth). The general rule of thumb is that the receiver will use atleast K measurements to sense the parameters of the K dominantreflectors.

More formally, let

={m₁, m₂, . . . } be a set of frequency elements (tones), where m_(i)takes values in the range [1, 2, . . . , N_(f)]. Note that the size of

, denoted by |

|, depends on different parameters such as the number of antennas L, thenumber of reflectors K, their delay profile and their angle of arrival(AOA) to the receiver antennas. Typically, |

| is very small and in some cases, may even be equal one (a singletone!).

The transmission of a small set of known reference signals over thetones of

is received over the L receiver's antennas. Let us define by M=|

|·L the number of received reference values, corresponding to thistransmission. Using the second-order statistics, an interpolation MMSEfilter, C, can be computed from these M received elements to estimatethe channel over all the N elements. Let H_(M) be the channel responseover these M received elements. Then, C may be computed as

C=R _(HH) _(M) ·(R _(H) _(M) _(H) _(M) )⁻¹  (5)

where R_(HH) _(M) and R_(H) _(M) _(H) _(M) corresponds to decimatedversions of the full R_(HH) matrix (one or two of the dimensions isreduced to M).

The PCA approximation for the second-order statistics can be useful toreduce the complexity of such a computation. Let V_(M) be the matrix Vdecimated along its N dimension to M. Then, C can be expressed as

C=V·D·V _(M)·(V _(M) ·D·V _(M))⁻¹ =V·(V _(M))⁻¹  (6)

The last computation requires inverting V_(M), which has the dimensionsof M×K. However, by algebraic manipulation, the inverted matrix may beeven further reduced to a size of K×K

C=V·(V _(M) *·V _(M))⁻¹ ·V _(M)*  (7)

Finally, the full channel response is obtained by computing

H=C·H _(M)  (8)

1.C. Estimating the Sufficient Number of Resource Elements

The receiver may use different methods, to estimate the required numberof (time-frequency) resource elements in M, to provide a sufficientquality of channel estimation. One method to do this, is to compute theerror covariance matrix

R _(E) =V·D·V*−C·V _(M) ·D·V*=(V−C·V _(M))·D·V*  (9)

and use it to compare the mean energy along the diagonal to a threshold

$\begin{matrix}{d_{R_{E}} = {{diag}( R_{E} )}} & (10) \\{\overset{\_}{R_{E}}\overset{\Delta}{=}{{\frac{1}{N}{\sum_{i = 1}^{N}{❘{d_{R_{E}}(i)}❘}^{2}}} < {TH}}} & (11)\end{matrix}$

This criterion may be used to determine whether the number of elementsin

is sufficient. Afterwards, the receiver may instruct the transmitter toadjust the number of required reference signals to transmit.

To illustrate this concept, an extreme case of channels with manyreflectors and small angle-of-arrivals was examined with differentnumber of tones allocated for reference signals. White Gaussian noisewas added to the received signal, creating a signal-to-noise (SNR) ratioof 60 dB. The graph in FIG. 4, shows for a varying number of tones withreference signals (1 to 5), the computed values of the average errorcovariance and the average estimation SNR (inverse of mean-square-errorcompared to the true channel). The computed average error covariance,R_(E), can be roughly compared to a threshold of 235 dB, afterwards thenoise contribution of the estimated channel is smaller than that of theAWGN. Therefore, for this example, it would be recommended to use 5tones with reference signals.

1.D. Embodiments for Prediction in FDD/TDD Networks

The methods described in the previous sections may also be applieddirectly for efficient channel prediction. Channel prediction mayinclude a training phase, where reference signals are transmitted overtwo different time-frequency resources sets and the second-orderstatistics is computed for the corresponding channels. These sets mayinclude two non-overlapping frequency bands, two different timeinstances, or a combination of both. For example, in FDD networks,downlink and uplink transmissions use different frequency bands, andchannel characteristics over one of downlink/uplink channel could bepredicted based on observed channel estimates over the other (uplink ordownlink) channel.

Let h_(k,1) represent a column vector of the channel response overN_(f1) frequency tones and L antennas, across a first frequency band andat time instance T_(k) (first set) and h_(k,2) as a column vector of thechannel response over N_(f2) frequency tones and L antennas, across asecond frequency band and at time instance T_(k)+Δ (second set). Amatrix H can be formulated as the concatenation of these column vectorsfrom multiple training sessions and compute the covariance matrixR_(HH).

$\begin{matrix} {{{H = \lbrack {\,_{h_{1,2}}^{h_{1,1}}❘_{h_{2,2}}^{h_{2,1}}} }❘}\ldots ❘_{h_{N_{SOS},2}}^{h_{N_{SOS},1}}} \rbrack & (12) \\{R_{HH} = {H \cdot H^{*}}} & (13)\end{matrix}$

The procedure described in previous sections, may be fully applied hereas well. PCA may be applied to R_(HH) for K dominant reflectors. In theoperational phase, the transmitter may send a small set of referencesignals, m, over part of the first time-frequency set and use thecomputed interpolation filter

C=R _(HH) _(M) ·(R _(H) _(M) _(H) _(M) )⁻¹  (14)

H=C·H _(M)  (15)

to estimate the channel across the entire second time-frequency set.

For example, after a training session, multiple user devices aretransmitting their small set of resources to the base station, where itis used for predicting future channels in a different band (FDD) or inthe same band (TDD). The base station uses the predicted channels tocompute a precoded transmission back to the users.

This method may also be applied for non-direct prediction techniques.For example, as a first step, the second-order statistics of the channelmay be computed from full-band reference signals. Then, small sets ofreference signals at both time-frequency sets are transmitted. Thereceived signals in each set, consisting of M elements may be furthercompressed to K elements by computing

H _(K)=(V _(M))⁻¹ ·H _(M)  (16)

Then, the covariance matrices of the two sets and the cross-covariancematrix may be computed along with an interpolation filter from one setto the other

R ⁽¹⁾ =H _(K) ⁽¹⁾·(H _(K) ⁽¹⁾)*  (17)

R ^((2.1)) =H _(K) ⁽²⁾·(H _(K) ⁽¹⁾)*  (18)

C ^((2.1)) =R ^((2.1))·(R ⁽¹⁾)⁻¹  (19)

Finally, the channel in the second set is predicted from the channel inthe first set

H _(K) ⁽²⁾ =C ^((2.1)) ·H _(K) ⁽¹⁾  (20)

and computed over all the N elements

H=V·H _(K) ⁽²⁾  (21)

1.E. Channel Responses and Channel Reciprocity

There are two main differences between the channel responses atterminals A and B and the channel response of the wireless channelreflectors:

1. The channel response of the wireless channel reflectors in atime-division duplex (TDD) system is reciprocal whereas the channelresponse of the terminals is not.

2. The channel response of the wireless channel reflectors may changerapidly (e.g., in 1-10 milliseconds, depending on the Doppler of thereflectors and terminals), but the channel response of the terminalschanges slowly, mostly with temperature.

There are several methods for obtaining the complete channel responsefrom terminal A to B described in the literature. For example, anexplicit method would be to send known reference signals from terminal Ato B and have terminal B transmit back the values of the receivedreference signals to terminal A. This is often referred to as explicitfeedback. However, each value must be represented with multiple bits,and in a system where terminal A has many antennas, there are many userterminals and significant Doppler effects causing the propagationchannel to change rapidly, the amount of information that needs to betransmitted can severely reduce the overall system efficiency. In theextreme case with high levels of Doppler, it is simply not possible tofeedback all the required Channel State Information (CSI) quicklyenough, resulting in stale CSI and suboptimal precoding.

Instead, a TDD system can use an approach known as “reciprocitycalibration” to obtain the relationship between the non-reciprocal partsof the channel response in both transmission directions: the AB (from Ato B) and the BA (from B to A). Terminal B first transmits knownreference signals that allow terminal A to compute the AB channelresponse. Using knowledge of the non-reciprocal relationship, terminal Acan adjust the BA channel response to make it suitable for precoding atransmission back to terminal B.

More formally, for a multi-carrier TDD system that uses multi-carriermodulation, where the channel can be described as a complex value in thefrequency domain for a specific sub-carrier (tone), the three componentsof the AB channel response can be denoted as H_(A) ^(TX), H^(CH) andH_(B) ^(RX). Similarly, the three components of the BA channel responseare H_(B) ^(TX), H^(CH) and H_(A) ^(RX). The overall downlink (AB)channel response is

H _(AB) =H _(A) ^(TX) ·H ^(CH) ·H _(B) ^(RX)  (22)

and the overall uplink (BA) channel response is

H _(BA) =H _(B) ^(TX) ·H ^(CH) ·H _(A) ^(RX)  (23)

From H_(AB) and H_(BA), the reciprocity calibration factor can bewritten as

$\begin{matrix}{\alpha = \frac{H_{A}^{TX} \cdot H_{B}^{RX}}{H_{B}^{TX} \cdot H_{A}^{RX}}} & (24)\end{matrix}$

Therefore, if H_(BA) is known at terminal A, it can computeH_(AB)=αH_(BA). The question that remains is how to obtain α. Note thatfor the multi-carrier system, the above Equations (22) to (24) willprovide reciprocity calibration values and channel responses on a persub-carrier basis for sub-carriers on which reference signals aretransmitted.

Different methods exist within the literature for computing thereciprocity calibration factor. The most straight forward of these is toutilize explicit feedback as described above, but only feed back H_(AB)when a is re-calculated. Since the transmitter and receiver channelresponses change relatively slowly, the rate of feedback is typically inthe order of minutes and thus represents negligible overhead for amodest number of terminals and antennas. However, when the number ofantennas in terminal A and the number of CPEs (terminal B) is large, ascan be the case in a massive multiple-input multiple-output (MIMO)system with many subscribers, the feedback overhead can consume aconsiderable portion of the system capacity.

Another approach is to have terminal A transmit reference signalsbetween its own antennas and calculate calibration factors for onlyH_(A) ^(TX) and H_(A) ^(RX). That is, obtain:

$\begin{matrix}{\alpha_{A} = \frac{H_{A}^{TX}}{H_{A}^{RX}}} & (25)\end{matrix}$

which results in

{tilde over (H)} _(AB)=α_(A) ·H _(BA) =H _(A) ^(TX) ·H _(B) ^(TX) ·H^(CH)  (26)

Terminal A will then precode one reference symbol using {tilde over(H)}_(AB) that terminal B can use to remove its H_(B) ^(TX) and H_(B)^(RX) contributions from all subsequent precoded transmissions. Thistechnique may be called relative calibration. Whilst this approachentirely removes the need for feedback of H_(BA), the need for terminalA to transmit to itself during a calibration procedure and then to CPEsthat could be located many hundreds of meters or even kilometers awaycan create dynamic range challenges. It is typically desirable to usethe same hardware gain stages in the transmit chain when calibrating asthose used for transmission, since having to switch gain stages betweencalibration and transmission can change the nature of H_(A) ^(TX) andH_(A) ^(RX).

This document describes, amongst other embodiments and approaches, a newapproach for computing the reciprocity calibration factor that avoidsthe dynamic range concern of relative calibration whilst maintaininghigh levels of efficiency when scaling to a larger number of antennasand terminals. As described herein, the reference signals transmittedfor calibration and at the same power level as typical signaltransmissions, and hence are better suited to capture and calibrate thedistortions introduced by transmit/receive circuitry.

1.F. Reciprocity Calibration Via Receiver-Side Inversion

Let Terminal A transmit known reference signals over a subset ofmulti-carrier tones and P be a specific reference signal at one of thesetones. For example, Terminal A may use every Mth subcarrier forreference signal transmission, where M is an integer. For example, M maybe 8 or 16 in practical systems. Terminal B receives

Y _(B) =H _(AB) ·P+W  (27)

where W is additive white Gaussian noise with zero mean and variance N₀.Note that the above equation is a scalar equation because the equationrepresents the received signal at a single subcarrier. For eachsubcarrier on which a reference is transmitted, there will be one suchequation. Terminal B estimates H_(AB) from Y_(B) and inverts it. Toavoid singularities and cope with a large dynamic range, regularizedzero forcing may be used to compute the inversion:

$\begin{matrix}{{\overset{\sim}{H}}_{AB}^{- 1} = {\frac{H_{AB}^{*}}{{H_{AB}^{*} \cdot H_{AB}} + N_{0}} \approx H_{AB}^{- 1}}} & (28)\end{matrix}$

Terminal B then transmits {tilde over (H)}_(AB) ⁻¹ back to terminal Aover the same tone. This transmission should quickly follow the firstone—especially in the presence of Doppler—to ensure H^(CH) remainsrelatively constant. Terminal A then receives

Y _(B) =H _(BA) ·{tilde over (H)} _(AB) ⁻¹ +W  (29)

Ignoring the noise term, which may be averaged out over multipletransmissions, it can be seen that Y_(B) is the inverse of thereciprocity calibration factor:

$\begin{matrix}{{Y_{B} \approx \frac{H_{B}^{TX} \cdot H^{CH} \cdot H_{A}^{RX}}{H_{A}^{TX} \cdot H^{CH} \cdot H_{B}^{RX}}} = {\frac{H_{B}^{TX} \cdot H_{A}^{RX}}{H_{A}^{TX} \cdot H_{B}^{RX}} = \alpha^{- 1}}} & (30)\end{matrix}$

Since these are scalar values, the inversion processing is for bothH_(AB) and Y_(B) is straightforward. Here, the inverse reciprocitycalibration factor represents a ratio of circuitry channel from TerminalB to Terminal A, and a circuitry channel from Terminal A to Terminal B.

In multi-carrier systems, the above-described procedure may be repeatedover multiple tones and the result interpolated to yield the full set ofcalibration factors over the bandwidth of interest. This full set may beobtained, for example, by averaging or interpolating the calibrationfactors are the subcarriers at which reference signals were transmitted.Since the Tx and Rx contributions of both terminal A and B will berelatively flat across frequency, it should be possible to use a sparsesubgrid of tones with the appropriate interpolation to obtain anaccurate level of calibration.

The results of the channel estimation as above may be combined withchannel estimation of the H^(CH) channel to obtain an estimate of theoverall channel H_(AB) and H_(BA).

2. Wireless Precoded Communication for FDD Systems

A wireless system, with a multi-antenna base-station and multiple userantennas, is shown in FIG. 5. Each transmission from a user antenna toone of the base-station antennas (or vice versa), experiences adifferent channel response (assuming the antennas are physicallyseparated enough). For efficient communication, the base-stationimproves the users' received Signal-to-Interference-Noise-Ratio (SINR)by means of precoding. However, to precode, the base-station needs tohave an accurate estimation of the downlink channels to the users duringthe transmission time.

In FDD systems, two directions of transmissions may use two different(typically non-overlapping) frequency bands. These transmissions mayinclude, for example, transmissions from a network-side node such as abase station or an access point to multiple user devices, often calledthe downlink direction, and transmissions from the multiple user devicesto the network-side node, often called the upstream or uplink direction.The various embodiments described in the present document, and theattached appendices, perform signal processing to improve communicationby, for example, estimating second order statistics of a wirelesschannel and using the estimate to perform pre-coding in an FDD system.

In the following subsections, an efficient system and a method forpredicting the downlink channel for precoding in an FDD system isdescribed.

2.A. Second-Order Statistics Training

For simplicity, this section focuses on a single user antenna and the Lbase-station antennas. This can be easily extended to any number ofusers. The setup of the system is shown in FIG. 6. The base-stationpredicts from the uplink channel response, the downlink channel responsein a different frequency band and N_(latency) subframes later.

To achieve this, the system preforms a preliminary training phase,consisting of multiple sessions, where in each session i=1, 2, . . . ,N_(training), the following steps are taken:

-   -   At subframe n, the user equipment transmits reference signals        (RS) in the uplink. The base-station receives them and estimate        the uplink channel H_(UL) ^((i)) over the L base-station        antennas.    -   At subframe n+N_(latency), the base-station transmits reference        signals in the downlink from all its antennas. The user        equipment receives it and sends it back as uplink data in a        later subframe. The base-station computes the downlink channel        estimation for it, H_(DL) ^((i)). In a different implementation,        it is possible that the UE will compute the channel estimation        and send it to the base-station as uplink data.    -   The base-station computes the second-order statistics

R _(UL) ^((i)) =H _(UL) ^((i))·(H _(UL) ^((i)))^(H)

R _(DL,UL) ^((i)) =H _(DL) ^((i))·(H _(UL) ^((i)))^(H)

R _(DL) ^((i)) =H _(DL) ^((i))·(H _(DL) ^((i)))^(H)

Herein, (⋅)^(H) is the Hermitian operator. For the case that the channelhas non-zero-mean, both the mean and the covariance matrix should bedetermined.

When the training sessions are completed, the base-station averages outthe second order statistics:

$\begin{matrix}{R_{UL} = {\frac{1}{N_{training}}{\sum_{i = 1}^{N_{training}}R_{UL}^{(i)}}}} \\{R_{{DL},{UL}} = {\frac{1}{N_{training}}{\sum_{i = 1}^{N_{training}}R_{{DL},{UL}}^{(i)}}}} \\{R_{DL} = {\frac{1}{N_{training}}{\sum_{i = 1}^{N_{training}}R_{DL}^{(i)}}}}\end{matrix}$

Then, it computes the prediction filter and the covariance of theestimation error:

C _(prediction) =R _(DL,UL)·(R _(UL))⁻¹

R _(E) =R _(DL) −C _(prediction)·(R _(DL,UL))^(H)

The inversion of R_(UL) may be approximated using principal componentanalysis techniques. We compute {λ}, the K most dominant eigenvalues ofR_(UL), arranged in a diagonal matrix D=diag(λ₁, λ₂, . . . , λ_(K)) andtheir corresponding eigenvectors matrix V. Typically, K will be in theorder of the number of reflectors along the wireless path. Thecovariance matrix can then be approximated by R_(UL)≈V·D·(V)^(H) and theinverse as R_(UL) ⁻¹≈V·D⁻¹·(V)^(H).

Note, that there is a limited number of training sessions and that theymay be done at a very low rate (such as one every second) and thereforewill not overload the system too much.

To accommodate for possible future changes in the channel response, thesecond-order statistics may be updated later, after the training phaseis completed. It may be recomputed from scratch by initializing againnew N_(training) sessions, or by gradually updating the existingstatistics.

The interval at which the training step is to be repeated depends on thestationarity time of the channel, i.e., the time during which thesecond-order statistics stay approximately constant. This time can bechosen either to be a system-determined constant, or can be adapted tothe environment. Either the base-station or the users can detect changesin the second-order statistics of the channel and initiate a newtraining phase. In another embodiment, the base-station may use thefrequency of retransmission requests from the users to detect changes inthe channel, and restart the process of computing the second-orderstatistics of the channel.

2.B. Scheduling a Downlink Precoded Transmission

For each subframe with a precoded downlink transmission, thebase-station should schedule all the users of that transmission to senduplink reference signals N_(latency) subframes before. The base-stationwill estimate the uplink channel responses and use it to predict thedesired downlink channel responses

H _(DL) =C _(prediction) ·H _(UL)

Then, the downlink channel response H_(DL) and the prediction errorcovariance R_(E) will be used for the computation of the precoder.

3. Efficient Channel Estimation Using Second-Order Statistics

In some embodiments, channel knowledge is typically acquired bytransmitting known reference signals (pilots) and interpolating them atthe receiver over the entire bandwidth and time of interest. Typically,the density of the pilots depends on characteristics of the channel.Higher delay spreads require more dense pilots along frequency andhigher Doppler spreads require more dense pilots along time. However,the pilots are typically required to cover the entire bandwidth ofinterest and, in some cases, also the entire time interval of interest.

Herein, a method based on the computation of the second-order statisticsof the channel, where after a training phase, the channel can beestimated over a large bandwidth from reference signals in a muchsmaller bandwidth is proposed. Even more, the channel can also bepredicted over a future time interval.

3.A. Second-Order Statistics Training for Channel Estimation

Similar to the embodiments described in Section 2.A, a system preforms apreliminary training phase, consisting of multiple sessions, where ineach session i=1, 2, . . . , N_(training), the following steps aretaken:

-   -   The transmitter sends reference signals to the receiver. We        partition the entire bandwidth of interest into two parts BW₁        and BW₂, as shown in FIG. 7, where typically the size of BW₁        will be smaller or equal to BW₂. Note, that these two parts do        not have to from a continuous bandwidth. The transmitter may        send reference signals at both parts at the same time interval        (FIG. 8) or at different time intervals (FIG. 9).    -   The receiver receives the reference signals and estimates the        channel over their associated bandwidth, resulting in channel        responses H₁ ^((i)) and H₂ ^((i)).    -   The receiver computes the second-order statistics of these two        parts:

R ₁ ^((i)) =H ₁ ^((i))·(H ₁ ^((i)))^(H)

R _(2,1) ^((i)) =H ₂ ^((i))·(H ₁ ^((i)))^(H)

R ₂ ^((i)) =H ₂ ^((i))·(H ₂ ^((i)))^(H)

Herein, (⋅)^(H) is the Hermitian operator. For the case that the channelhas non-zero-mean, both the mean and the covariance matrix should bedetermined, as is further discussed in the attached appendices. When thetraining sessions are completed, the base-station averages out thesecond order statistics:

$\begin{matrix}{R_{1} = {\frac{1}{N_{training}}{\sum_{i = 1}^{N_{training}}R_{1}^{(i)}}}} \\{R_{2,1} = {\frac{1}{N_{training}}{\sum_{i = 1}^{N_{training}}R_{2,1}^{(i)}}}} \\{R_{2} = {\frac{1}{N_{training}}{\sum_{i = 1}^{N_{training}}R_{2}^{(i)}}}}\end{matrix}$

Then, the receiver computes a prediction filter:

C _(prediction) =R _(2,1)·(R ₁)⁻¹

and if needed, also the covariance of the estimation error

R _(E) =R ₂ −C _(prediction)·(R _(2.1))^(H)

The inversion of R₁ may be approximated using principal componentanalysis techniques. We compute {λ}, the K most dominant eigenvalues ofR₁, arranged in a diagonal matrix D=diag(λ₁, λ₂, . . . , λ_(K)) andtheir corresponding eigenvectors matrix V. Typically, K will be in theorder of the number of reflectors along the wireless path. Thecovariance matrix can then be approximated by R₁≈V·D·(V)^(H) and theinverse as R₁ ⁻¹≈V·D⁻¹·(V)^(H).

Note, that there is a limited number of training sessions and that theymay be done at a very low rate (such as one every second) and thereforewill not overload the system too much.

To accommodate for possible future changes in the channel response, thesecond-order statistics may be updated later, after the training phaseis completed. It may be recomputed from scratch by initializing againnew N_(training) sessions, or by gradually updating the existingstatistics.

The interval at which the training step is to be repeated depends on thestationarity time of the channel, i.e., the time during which thesecond-order statistics stay approximately constant. This time can bechosen either to be a system-determined constant, or can be adapted tothe environment. Either the base-station or the users can detect changesin the second-order statistics of the channel and initiate a newtraining phase. In another embodiment, the base-station may use thefrequency of retransmission requests from the users to detect changes inthe channel, and restart the process of computing the second-orderstatistics of the channel.

3.B. Embodiments for Efficient Channel Estimation

After the training phase is completed, the transmitter may only sendreference signals corresponding to BW₁. The receiver, estimated thechannel response H₁ and use it to compute (and predict) and channelresponse H₂ over BW₂ using the prediction filter:

H ₂ =C _(prediction) ·H ₁

FIGS. 10 and 11 describe example two prediction scenarios (same timeinterval and future time interval, respectively).

4. Reciprocal Calibration for Reverse Channel Estimation

In the operation of the wireless systems described in this document, n−1signals, intended for n−1 individual UEs, will act as interference forthe target UE. A transmit pre-coder cancels the interference generatedat the target UE by the n−1 un-intended signals meant for other UEs. Tobuild a pre-coder, down link channel state information (CSI) is used.

In an extrinsic beamforming technique, CSI is fed back from the UE to BSthrough a feedback up-link channel. However, considerable amount of dataBW is used for this, thus affecting the overall system throughputefficiency.

For Time Division Duplex (TDD) systems, the physical channel in the air(sometimes called the radio channel) is reciprocal within the channelcoherence time. i.e., the up-link (UE to BS) and down-link (BS to UE)are identical (in SISO (transpose in MIMO). However, when thetransceiver front-end (FE) hardware is also taken into account, channelreciprocity no longer holds. This is due to the non-symmetriccharacteristics of the RF hardware. It includes PA non-linearity, RFchain crosstalk, phase noise, LNA non-linearity and noise figure,carrier and clock drifts etc.

In some embodiments, a calibration mechanism can be designed tocalibrate for the nonreciprocal components of the wireless link suchthat embodiments can estimate the down-link by observing the up-linkwith the help of these calibration coefficients. If this is feasible, noCSI feedback is necessary (as in the case of extrinsic beam forming),thus improving the overall system throughput efficiency. The associatedbeamforming is also sometimes called intrinsic beamforming.

FIG. 12 shows an example block diagram of a communication channel withreciprocity. The composite wireless channel from A to B may berepresented as: Ĥ_(A,B)=C_(RX,B). H_(A,B). C_(TX,A). For the reciprocalchannel, it may be assumed that H_(AB)=λH_(BA) ^(T) for a complex scalar(λ).

In the case of a non-reciprocal channel, with analog and RF components,Non-reciprocal analog and RF components: C_(TX, A), C_(RX, A),C_(RX, B), C_(TX, B), ideally for simplicity, it is beneficial if eachmatrix is a diagonal matrix. Such an embodiment may also use a designthat minimizes the coupling between Tx and Rx paths. Similarly, thecomposite channel from B to A is given byĤ_(B,A)=C_(RX, A)·H_(B,A)·C_(TX,B).

If all the C matrices can be estimated a priori, the BS to UE channelcan be estimated from the UE to BS channel. In such a case, feeding backchannel state information for transmit beamforming may not be needed,thereby making the upstream bandwidth available for data instead ofhaving to transmit channel state information. Estimation of the Cmatrices may also improve system efficiency. In some embodimentsdisclosed herein, the reciprocity calibration may be performed bycalibrating Tx and Rx of the BS and UE side during a startup or apre-designated time. The diagonal matrices C_(TX, A), C_(RX, A),C_(RX, B), C_(TX, B) may be estimated. These matrices may bere-estimated and updated periodically. The rate of change of the Cmatrices will typically be slow and may be related to factors such asthe operating temperature of the electronics used for Tx and Rx.

4.A. Notation

In the description herein, h_(a1a2) denotes the channel from transmitter(TX) a1 to receiver (RX) a2. This notation is different from theconventional MIMO channel notation. In the conventional methods, thiswill be denoted as h_(a2a1)). Also, conjugate of a complex quantity isrepresented with a *, e.g., conj(h)=h*.

4.B. Downlink Channel Estimation from Uplink Channel and CalibrationCoefficients

While the disclosed techniques are more generally applicable, for theease of explanation, the following assumptions are made:

[1] cross talk between the TX-TX, RX-RX and TX-RX RF chains isnegligible

[2] Antenna mutual coupling in negligible

[3] TX and RX at BS are working with clocks generated from the same PLLso that carrier and symbol clocks are synchronized in wirelesstransmissions among transceivers at A or B (and not necessarily betweenA and B). Here, “A” and “B” represent communication devices at two endsof a communication medium. For example, A may refer to a base station(or user equipment) and correspondingly B may refer to a user equipment(or base station). As another example, A may refer to a hub and B mayrefer to a remote station. Without loss of generality, the channel fromA to B may be called the downlink (DL) channel and the channel from B toA may be called the uplink (UL) channel. See, e.g., FIG. 13 and FIG. 14.

[4] same assumptions as above for UE.

Inventors' measurement on some existing equipment has verified that a)the coupling between different RF paths is typically of the order of −30dB. A careful design of the RF front end can ensure even lesser levelsof cross talk. b) The isolation between the cross polarizations of theantenna is of the order of 15 to 20 dB. This means that if a signal of xdB power is sent on the vertical polarization of a cross polarizedantenna, an image with (x−15) dB power will appear on the horizontalpolarization. This isolation cannot be improved much even under improvedantenna design. So, for the below calibration mechanism to workproperly, embodiments should either use i) antenna with singlepolarization is used or ii) if dual polarized antenna are used, takecare that simultaneous transmission on both the polarizations is neverhappening.

However, these assumptions can be relaxed, as described herein, andmodifications in the below described calibration algorithm will bepresented as well. If dual polarized antenna is the design choice,modifications to the disclosed algorithm as described herein could beused in some embodiments.

Some embodiments of a calibration algorithm are described herein for a4×4 MIMO system. This is to keep the description simple and easy tocomprehend. The same mechanism can be generalized to systems with anynumber of BS and UE antenna.

With reference to FIG. 13 and FIG. 14, a four antenna system 1300 at Aand a four antenna system 1400 at B are depicted, respectively. Forexample, the configuration 1300 at A may represent a base station andthe configuration 1400 at B may represent a UE (or vice versa). Letĥ_(a1a2) denote the channel from TX a1 to RX a2. It is constituted by 2components a). The reciprocal radio channel from antenna a1 to antennaa2 and b) the non-reciprocal components at TX a1 and RX a2.Non-reciprocal components are captured in two memory-less complexscalars denoted by t_(a1) and r_(a2) corresponding to the TX and RXrespectively. In this patent document, modifications when delay isinvolved will be demonstrated as well.

Thus, ĥ_(a1a2) can be written as:

${{\hat{h}}_{a1a2} = {t_{a1} \cdot h_{a1a2} \cdot r_{a2}}}{{\hat{h}}_{a2a1},{{\hat{h}}_{a2a1} = {t_{a2} \cdot h_{a2a1} \cdot r_{a1}}}}{{\hat{h}}_{a1a2}{\hat{h}}_{a2a1}}{\begin{matrix}{c_{a1a2} = \frac{t_{a1} \cdot r_{a2}}{r_{a1} \cdot t_{a2}}} \\{= \frac{1}{c_{a2a1}}}\end{matrix}}$

The coefficient c_(a1a2) is referred to as the calibration coefficientfrom a1 to a2. Similarly, the calibration coefficient from a2 to a3,c_(a2a3), can be written as:

$\begin{matrix}{c_{a2a3} = \frac{t_{a2} \cdot r_{a3}}{r_{a2} \cdot t_{a3}}} \\{= {\frac{t_{a1} \cdot r_{a3}}{r_{a1} \cdot t_{a3}} \cdot \frac{r_{a1} \cdot t_{a2}}{t_{a1} \cdot r_{a2}}}} \\{= \frac{c_{a1a3}}{c_{a1a2}}}\end{matrix}$

From Eq. (34), it can be seen that the calibration coefficient betweenantenna a2 (TX) and antenna a3 (RX) can be written as the ratio of thecalibration coefficients of the reference antenna a1 to that of a3 anda2. Similar relations can be derived for TXs and RXs at B, depicted inconfiguration 1400 of FIG. 14. Equations for calibration coefficientsthat involve antennas from both sides A and B can be derived as below.

$\begin{matrix}{c_{a2b2} = \frac{t_{a2} \cdot r_{b2}}{t_{b2} \cdot r_{a2}}} \\{= {\frac{t_{a1} \cdot r_{b1}}{t_{b1} \cdot r_{a1}} \cdot \frac{r_{a1} \cdot t_{a2}}{t_{a1} \cdot r_{a2}} \cdot \frac{t_{b1} \cdot r_{b2}}{t_{b2} \cdot r_{b1}}}} \\{= {\frac{c_{a1b1}}{c_{a1a2}} \cdot c_{b1b2}}}\end{matrix}$

From Eq. (35), it should be clear that any wireless channel between Aand B can be written in terms of a) a calibration coefficient withrespect to a reference antenna at A, b) a calibration coefficient withrespect to a reference antenna at B, and c) a calibration coefficientbetween the same reference antennas at A and B.

Similarly, the downstream channels (BS-UE) can be represented in termsof the upstream channels (UE-BS) and calibration coefficients.

ĥ _(a1b1) =c _(a1b1) ·ĥ _(b1a1)

ĥ _(a1b2) =c _(a1b1) ·c _(b1b2) ·ĥ _(b2a1)(c _(a1b2) ·ĥ _(b2a1))

ĥ _(a1b3) =c _(a1b1) ·c _(b1b3) ·ĥ _(b3a1)(c _(a1b3) ·ĥ _(b3a1))

ĥ _(a1b4) =c _(a1b1) ·c _(b1b4) ·ĥ _(b4a1)

Similarly,

${{\hat{h}}_{a2b1} = {\frac{c_{a1b1}}{c_{a1a2}} \cdot {\hat{h}}_{b1a2}}}{{\hat{h}}_{a2b2} = {\frac{c_{a1b1}}{c_{a1a2}} \cdot c_{b1b2} \cdot {\hat{h}}_{b2a2}}}{{\hat{h}}_{a2b3} = {\frac{c_{a1b1}}{c_{a1a2}} \cdot c_{b1b3} \cdot {\hat{h}}_{b3a2}}}{{\hat{h}}_{a2b4} = {\frac{c_{a1b1}}{c_{a1a2}} \cdot c_{b1b4} \cdot {\hat{h}}_{b4a2}}}$

Furthermore,

${{\hat{h}}_{a3b1} = {\frac{c_{a1b1}}{c_{a1a3}} \cdot {\hat{h}}_{b1a3}}}{{\hat{h}}_{a3b2} = {\frac{c_{a1b1}}{c_{a1a3}} \cdot c_{b1b2} \cdot {\hat{h}}_{b2a3}}}{{\hat{h}}_{a3b3} = {\frac{c_{a1b1}}{c_{a1a3}} \cdot c_{b1b3} \cdot {\hat{h}}_{b3a3}}}{{\hat{h}}_{a3b4} = {\frac{c_{a1b1}}{c_{a1a3}} \cdot c_{b1b4} \cdot {\hat{h}}_{b4a3}}}$

And finally

${{\hat{h}}_{a4b1} = {\frac{c_{a1b1}}{c_{a1a4}} \cdot {\hat{h}}_{b1a4}}}{{\hat{h}}_{a4b2} = {\frac{c_{a1b1}}{c_{a1a4}} \cdot c_{b1b2} \cdot {\hat{h}}_{b2a4}}}{{\hat{h}}_{a4b3} = {\frac{c_{a1b1}}{c_{a1a4}} \cdot c_{b1b3} \cdot {\hat{h}}_{b3a4}}}{{\hat{h}}_{a4b4} = {\frac{c_{a1b1}}{c_{a1a4}} \cdot c_{b1b4} \cdot {\hat{h}}_{b4a4}}}$

Using the results from Eq. (36) to Eq (39) and denoting c_(a1b1) as ζ, acomplex constant, the downlink MIMO channel can be expressed in terms ofthe uplink MIMO channel using the following equation:

$\begin{bmatrix}{\hat{h}}_{a1b1} & {\hat{h}}_{a2b1} & {\hat{h}}_{a3b1} & {\hat{h}}_{a4b1} \\{\hat{h}}_{a1b2} & {\hat{h}}_{a2b2} & {\hat{h}}_{a3b2} & {\hat{h}}_{a4b2} \\{\hat{h}}_{a1b3} & {\hat{h}}_{a2b3} & {\hat{h}}_{a3b3} & {\hat{h}}_{a4b3} \\{\hat{h}}_{a1b4} & {\hat{h}}_{a2b4} & {\hat{h}}_{a3b4} & {\hat{h}}_{a4b4}\end{bmatrix} = {\zeta \cdot \begin{bmatrix}{\hat{h}}_{b1a1} & {\frac{1}{c_{a1a2}}{\hat{h}}_{b1a2}} & {\frac{1}{c_{a1a3}}{\hat{h}}_{b1a3}} & {\frac{1}{c_{a1a4}}{\hat{h}}_{b1a4}} \\{c_{b1b2}{\hat{h}}_{b2a1}} & {\frac{c_{b1b2}}{c_{a1a2}}{\hat{h}}_{b2a2}} & {\frac{c_{b1b2}}{c_{a1a3}}{\hat{h}}_{b2a3}} & {\frac{c_{b1b2}}{c_{a1a4}}{\hat{h}}_{b2a4}} \\{c_{b1b3}{\hat{h}}_{b3a1}} & {\frac{c_{b1b3}}{c_{a1a2}}{\hat{h}}_{b3a2}} & {\frac{c_{b1b3}}{c_{a1a3}}{\hat{h}}_{b3a3}} & {\frac{c_{b1b3}}{c_{a1a4}}{\hat{h}}_{b3a4}} \\{c_{b1b4}{\hat{h}}_{b4a1}} & {\frac{c_{b1b4}}{c_{a1a2}}{\hat{h}}_{b4a2}} & {\frac{c_{b1b4}}{c_{a1a3}}{\hat{h}}_{b4a3}} & {\frac{c_{b1b4}}{c_{a1a4}}{\hat{h}}_{b4a4}}\end{bmatrix}}$

The right hand side of Eq. (40a) can be further decomposed as:

$= {\zeta \cdot \begin{bmatrix}1 & 0 & 0 & 0 \\0 & c_{b1b2} & 0 & 0 \\0 & 0 & c_{b1b3} & 0 \\0 & 0 & 0 & c_{b1b4}\end{bmatrix} \cdot \lbrack \text{⁠}\begin{matrix}{\hat{h}}_{b1a1} & {\hat{h}}_{b1a2} & {\hat{h}}_{b1a3} & {\hat{h}}_{b1a4} \\{\hat{h}}_{b2a1} & {\hat{h}}_{b2a2} & {\hat{h}}_{b2a3} & {\hat{h}}_{b2a4} \\{\hat{h}}_{b3a1} & {\hat{h}}_{b3a2} & {\hat{h}}_{b3a3} & {\hat{h}}_{b3a4} \\{\hat{h}}_{b4a1} & {\hat{h}}_{b4a2} & {\hat{h}}_{b4a3} & {\hat{h}}_{b4a4}\end{matrix} \rbrack \cdot {\lbrack \text{⁠}\begin{matrix}1 & 0 & 0 & 0 \\0 & \frac{1}{c_{a1a2}} & 0 & 0 \\0 & 0 & \frac{1}{c_{a1a3}} & 0 \\0 & 0 & 0 & \frac{1}{c_{a1a4}}\end{matrix} \rbrack}}$

Eq. (40b) is of the form ζ. K_(B). H_(U). K_(A). Elements in thecalibration coefficient matrix K_(A) and K_(B) are obtained bycalibrations performed at A (BS) and B (UE) and later by transferring UEcoefficients to the BS. Note that calibration coefficient estimation atthe BS may involve transmission and reception of calibration signalsamong BS antennas (a.k.a. local calibration). Similarly, estimation ofcalibration coefficients at UE can be performed using local transmissionand reception of calibration signals among UE antennas.

In some embodiments, the TX and RX timing at a device (BS or UE) may beoperated from the same PLL. This eliminates the carrier and/or clockoffset impairments that is often associated with the detector at B forRF transmissions from A and vice versa. This is because A and B will bederiving all their internal clock frequencies, in general, from 2different PLLs, one at A and another one at B. If these impairments area part of the calibration coefficients, e.g., manifest themselves as atime varying phase rotation, then the coefficients will vary morefrequently due to time varying carrier or clock errors in addition toits own time variability. Since KA and K_(B) are obtained frommeasurements exclusively at BS or UE (and not using transmissions fromBS to UE or vice versa), they vary relatively slowly.

Local calibrations, thus, are generally stable and do not change muchover a period of several minutes (e.g., 30 minutes).

Eq. (40b) further reveals that the reverse MIMO channel can bemathematically modelled as the composition of a) a complex scalar, b)calibration coefficients at B, c) MIMO uplink channel transfer function,and d) calibration coefficients at A.

A pre-coder can be built at A (or at B), by acquiring calibrationcoefficients of the receiver side. The pre-coder implementation could bebased on any of several pre-coders available in the literature. Someexamples include: an MMSE pre-coder, a regularized MMSE precoder, a zeroforcing precoder, a Tomlinson-Harashima pre-coder, and so on. As will beappreciated by one of skill in the art, either linear pre-coders (firstthree examples above) or non-linear pre-coders (the last example above)may be used.

To illustrate this point, an example embodiment of a reciprocity basedzero-forcing (ZF) pre-coder using the data above (ref. Eq. 40b) isdescribed below. The configuration of this pre-coder is depicted in FIG.15. Note that, the same set of data can be used to design any type ofpre-coder.

A ZF pre-coder may have the following form.

${\text{?}W} = \frac{K_{A}^{- 1} \cdot H_{U}^{- 1} \cdot K_{B}^{- 1}}{E\{ {{{K_{A} \cdot H_{U}}{\cdot}_{B}}}_{F} \}}$._(F)? ?indicates text missing or illegible when filed

Some embodiments use the fact that K_(B) and K_(A) are typicallyslow-varying, so that their inverse can be implemented in software(instead of implementing in hardware). In some embodiments, W2 (H_(U) ⁻¹or inverse of H_(U)) may be fast varying and implemented in hardwarecircuits.

In some embodiments, W2 may be obtained as a by-product of the receiverEqualizer at the BS. For example BS equalizer often implements a variantof the H_(U) ⁻¹.

Therefore, using the techniques disclosed herein, some embodiments maymodel the downlink channel as a composition of the uplink channel andslow varying local coefficients. This enables to build a variety ofdifferent pre-coders with minimal-feedback overhead, and certain linearprecoders, use receiver equalizer computations performed at the basestation.

Estimation of ζ or c_(a1b1) involves calibrating across BS and UE. Forthe reasons discussed above, these coefficients can be frequentlyvarying and the estimation and feedback of these coefficients couldconsume a lot of bandwidth. Advantageously, a transmit-side pre-coder tocancel the multi-user interference can be designed without the knowledgeof ζ. It can be designed from the upstream channel measurements K_(A)and K_(B), as described in the present document.

Several estimation methods to determine the calibration coefficients aredescribed in the present document. When the number of antenna isrelatively small, the method described in Section 4.C may be used. Whena large number of antennas are involved, the method described in Section4.D may be used.

4.C. Estimation of Calibration Coefficients (c_(x1x2))—Method1—Iterative Algorithm

When the number of antenna is relatively small, such as 4 as in thisexample, an iterative algorithm can be used to compute the calibrationcoefficients. This, however, will entail a large amount of calibrationsin a massive MIMO scenario and may become impractical. In a massive MIMOscenario, estimation described in sec. III may be used.

Iteration 1:

c _(a1a3) =c _(a1a2) ·c _(a2a3)

c _(a1a4) =c _(a1a2) ·c _(a2a4)

Make an initial estimate of c_(a1a2), c_(a1a3), c_(a1a4) as describedherein. Also estimate c_(a2a3) and c_(a2a4). Using the latter, make analternative estimate of c_(a1a3) and c_(a1a4) (eq. 41 and 42). Fromthese two independent estimates of c_(a1a3) and c_(a1a4), make a refinedestimate of c_(a1a3) and c_(a1a4) (e.g., the arithmetic mean). A refinedestimate of c_(a1a2) can be obtained from Eq. (41) or (42)

A refined estimate of c_(a1a4) can be obtained (as described above),from a measurement of c_(a3a4) using Eq. (43). Using the updatedc_(a1a4), estimates of c_(a1a2) and c_(a1a3) can be updated.

More iterations, in the above manner, can be performed to further refinethe above coefficients.

The number of calibrations to be performed in this method is 4_(C2)=6.Number of calibrations will grow exponentially in N, the number ofantenna; hence not practical for large array of antenna.

4.D. Estimation of calibration coefficients (C_(x1x2))—Method 2—Largenumber of antenna

When the number of antenna is high, e.g., 64, Method 1 is notpractically feasible. This section describes a simple method for thesame.

The following equations have been previously seen derived.

ĥ_(a1a2) = c_(a1a2)ĥ_(a2a1) y_(a2) = ĥ_(a1a2)s_(a1) + n_(a2)y_(a1) = ĥ_(a2a1)s_(a2) + n_(a1)${\hat{h}}_{a1a2} = \frac{y_{a2} \cdot s_{a1}^{*}}{{❘s_{a1}❘}^{2}}$${\hat{h}}_{a2a1} = \frac{y_{a1} \cdot s_{a2}^{*}}{{❘s_{a2}❘}^{2}}$${CN}( {0,\frac{\sigma_{n}^{2}}{{❘s_{b}❘}^{2}}} )$$\frac{{\hat{h}}_{a1a2}}{{\hat{h}}_{a2a1}}$

For a 64 antenna system, this method would use a total of 64 calibrationsignal transmissions.

4.E. Estimation of Calibration Coefficients (C_(x1x2))—Method 3—UsingTotal Least Squares (TLS)

What is described below is an algorithm to estimate the calibrationcoefficients using the method of total least squares.

Refer to Method 2. Take k (say 4) such LS estimates and form thefollowing matrix

${H_{a1a2} = \lbrack {{\hat{h}}_{{a1a2},1},{{\hat{h}}_{{a1a2},2}\ldots{\hat{h}}_{{a1a2},4}}} \rbrack}{H_{a2a1} = \lbrack {{\hat{h}}_{{a2a1},1},{{\hat{h}}_{{a2a1},2}\ldots{\hat{h}}_{{a2a1},4}}} \rbrack}\lbrack {H_{a2a1},H_{a1a2}} \rbrack{c_{a1a2} = {{\underset{{\Delta H_{a1a2}},{\Delta H}_{a2a1}}{argmin}{❘{\Delta H_{a2a1}}❘}_{F}^{2}} + {❘{\Delta H_{a1a2}}❘}_{F}^{2}}}{{H_{a1a2} + {\Delta H_{a1a2}}} = {( {H_{a2a1} + {\Delta H_{a2a1}}} ) \cdot c_{a1a2}}}$

Solution for the above is obtained as below,

${D = {U\Sigma V^{H}}}{{\Sigma = {{diag}( {\sigma_{1},\sigma_{2}} )}},{\sigma_{1} > \sigma_{2}}}{\sigma s}{V = {{{\begin{bmatrix}v_{11} & v_{12} \\v_{21} & v_{22}\end{bmatrix}.\text{?}}v_{22}} \neq 0}}{{\text{?}\sigma_{1}} \neq \sigma_{2}}{{\hat{c}}_{{a1a2} - {opt}} = \frac{- v_{12}}{v_{22}}}{\text{?}h_{\cdot {(\ldots)}}}{\text{?}c_{\cdot {(\ldots)}}}{\text{?}h_{\cdot {(\ldots)}}}$?indicates text missing or illegible when filed

Note that three different methods for estimating the calibrationcoefficients are described above. During the implementation/prototypingphase, effectiveness of each method can be evaluated separately and theone with the best merit need to be selected for the finalimplementation. Alternatively, the decision may be made based on thenumber of antennas for which the calibration is performed. For example,method 1 may be suitable for up to 8 receive and/or transmit antennas,while methods 2 or 3 may be used for higher number of antennas.

4.F. Effect of Delays in the TX/RX Path

${t_{a1},t_{a2},r_{a1}}r_{a2}{\tau_{{ta}1},\tau_{{ta}2},\tau_{{ra}1},\tau_{{ra}2}}\begin{matrix}{c_{a1a2} = {\frac{❘{t_{a1} \cdot r_{a2}}❘}{❘{r_{a1} \cdot t_{a2}}❘} \cdot \frac{e^{{- j}2\pi f\tau_{{ra}2}} \cdot e^{{- j}2\pi f\tau_{{ta}1}}}{e^{{- j}2\pi f\tau_{{ra}1}} \cdot e^{{- j}2\pi f\tau_{{ta}2}}}}} \\{= {{❘c_{a1a2}❘} \cdot e^{{- j}2\pi f{\delta\tau}}}}\end{matrix}{{\delta\tau} = {( {\tau_{{ta}1} + \tau_{{ra}2}} ) - ( {\tau_{{ta}2} + \tau_{{ra}1}} )}}{\text{?}{\delta\tau}}{\text{?}{\delta\tau}}{\text{?}t_{a1}}{\text{?}t_{a1}}$?indicates text missing or illegible when filed

4.G. Calibration Coefficients when there is Mutual Coupling BetweenTX/TX, TX/RX Paths

Non-negligible coupling between different TX/RX path can exist due to a)Imperfect hardware design or more importantly b) if cross-polarizedantenna is used in the design. Modifications to the calibrationalgorithm under these conditions is described below. It is assumed thatthere is no coupling between the TX and RX paths.

Referring to FIG. 13 and FIG. 14, assume that the receive paths havemutual coupling between them. For e.g., r_(a1a2) is the mutual couplingbetween receive paths r₁ and r₂. Similarly r_(a1a3) refers to thecoupling between the receive paths r₁ and r₃. Similar explanationapplies to mutual coupling between the transmit paths. Note thatr_(a1a1) refers to r_(a1) of the Section 4.B.

Assume that antenna a1 transmits a calibration sequence, sa1. It hasbeen received by receive antenna a₂, a₃ and a₄. Similar to theformulation in Sections 4.B. and 4.C., the following expressions can bederived.

${\begin{bmatrix}y_{a2}^{\prime} \\y_{a3}^{\prime} \\y_{a4}^{\prime}\end{bmatrix} = {{{\begin{bmatrix}r_{a2a2} & r_{a2a3} & r_{a2a4} \\r_{a3a2} & r_{a3a3} & r_{a3a4} \\r_{a4a2} & r_{a4a3} & r_{a4a4}\end{bmatrix}\begin{bmatrix}h_{a1a2} \\h_{a1a3} \\h_{a1a4}\end{bmatrix}}{t_{a1} \cdot s_{a1}}} + \begin{bmatrix}n_{a2} \\n_{a3} \\n_{a4}\end{bmatrix}}}{\begin{bmatrix}y_{a1}^{''} \\y_{a3}^{''} \\y_{a4}^{''}\end{bmatrix} = {{{\begin{bmatrix}r_{a1a1} & r_{a1a3} & r_{a1a4} \\r_{a3a1} & r_{a3a3} & r_{a3a4} \\r_{a4a1} & r_{a4a3} & r_{a4a4}\end{bmatrix}\begin{bmatrix}h_{a2a1} \\h_{a2a3} \\h_{a2a4}\end{bmatrix}}{t_{a2} \cdot s_{a2}}} + \begin{bmatrix}n_{a1} \\n_{a3} \\n_{a4}\end{bmatrix}}}{\begin{bmatrix}y_{a1}^{\prime\prime\prime} \\y_{a2}^{\prime\prime\prime} \\y_{a4}^{\prime\prime\prime}\end{bmatrix} = {{{\begin{bmatrix}r_{a1a1} & r_{a1a2} & r_{a1a4} \\r_{a2a1} & r_{a2a2} & r_{a2a4} \\r_{a4a1} & r_{a4a2} & r_{a4a4}\end{bmatrix}\begin{bmatrix}h_{a3a1} \\h_{a3a2} \\h_{a3a4}\end{bmatrix}}{t_{a3} \cdot s_{a3}}} + \begin{bmatrix}n_{a1} \\n_{a2} \\n_{a4}\end{bmatrix}}}{\begin{bmatrix}y_{a1}^{\prime\prime\prime\prime} \\y_{a2}^{\prime\prime\prime\prime} \\y_{a3}^{\prime\prime\prime\prime}\end{bmatrix} = {{{\begin{bmatrix}r_{a1a1} & r_{a1a2} & r_{a1a3} \\r_{a2a1} & r_{a2a2} & r_{a2a3} \\r_{a3a1} & r_{a3a2} & r_{a3a3}\end{bmatrix}\begin{bmatrix}h_{a4a1} \\h_{a4a2} \\h_{a4a3}\end{bmatrix}}{t_{a4} \cdot s_{a4}}} + \begin{bmatrix}n_{a1} \\n_{a2} \\n_{a3}\end{bmatrix}}}{\text{?}{\hat{h}}_{a1a2}}{\text{?}h_{a1a3}}{\text{?}{\hat{h}}_{a1a4}}{\text{?}{\hat{h}}_{a1a2}}{\text{?}h_{a1a2}}{\text{?}{\hat{h}}_{a2a1}}{\text{?}{\hat{h}}_{a3a1}}{\text{?}{\hat{h}}_{a4a1}}{\begin{bmatrix}{\hat{h}}_{a1a2} \\{\hat{h}}_{a1a3} \\{\hat{h}}_{a1a4}\end{bmatrix} = {t_{a1} \cdot {\begin{bmatrix}r_{a2a2} & r_{a2a3} & r_{a2a4} \\r_{a3a2} & r_{a3a3} & r_{a3a4} \\r_{a4a2} & r_{a4a3} & r_{a4a4}\end{bmatrix}\begin{bmatrix}h_{a1a2} \\h_{a1a3} \\h_{a1a4}\end{bmatrix}}}}{\begin{bmatrix}{\hat{h}}_{a2a1} \\{\hat{h}}_{a3a1} \\{\hat{h}}_{a4a1}\end{bmatrix} = {\begin{bmatrix}{r_{a1a1} \cdot t_{a2}} & {r_{a1a3} \cdot t_{a2}} & {r_{a1a4} \cdot t_{a2}} \\{r_{a1a1} \cdot t_{a3}} & {r_{a1a2} \cdot t_{a3}} & {r_{a1a4} \cdot t_{a3}} \\{r_{a1a1} \cdot t_{a4}} & {r_{a1a2} \cdot t_{a4}} & {r_{a1a3} \cdot t_{a4}}\end{bmatrix}\begin{bmatrix}h_{a2a1} \\h_{a3a1} \\h_{a4a1}\end{bmatrix}}}$ ?indicates text missing or illegible when filed

From Eq. 55 and 56, we can write the following

$\begin{bmatrix}{\hat{h}}_{a2a1} \\{\hat{h}}_{a3a1} \\{\hat{h}}_{a4a1}\end{bmatrix} = {\underset{C}{\underset{︸}{\frac{1}{t_{a1}} \cdot {\begin{bmatrix}{r_{a1a1} \cdot t_{a2}} & {r_{a1a3} \cdot t_{a2}} & {r_{a1a4} \cdot t_{a2}} \\{r_{a1a1} \cdot t_{a3}} & {r_{a1a2} \cdot t_{a3}} & {r_{a1a4} \cdot t_{a3}} \\{r_{a1a1} \cdot t_{a4}} & {r_{a1a2} \cdot t_{a4}} & {r_{a1a3} \cdot t_{a4}}\end{bmatrix}\begin{bmatrix}r_{a2a2} & r_{a2a3} & r_{a2a4} \\r_{a3a2} & r_{a3a3} & r_{a3a4} \\r_{a4a2} & r_{a4a3} & r_{a4a4}\end{bmatrix}}^{- 1}}}\begin{bmatrix}{\hat{h}}_{a1a2} \\{\hat{h}}_{a1a3} \\{\hat{h}}_{a1a4}\end{bmatrix}}$

where C is the calibration coefficient to be estimated. It isinteresting to note that when the mutual coupling coefficients are setto 0, the above equation gives raise to eq. 33. Similarly ĥ_(a2a3)ĥ_(a2a4) and ĥ_(a3a4) can be expressed in terms of the product of acalibration matrix and ĥ_(a3a2), ĥ_(a4a2) and ĥ_(a4a3).

These calibration coefficients can be computed using Total Least Squaresmethod (method 3).

Note that there is no mutual coupling between sides A and B. Thisenables embodiments to write downstream link exactly as in Eq. 40c.However, in this case, K_(A) and K_(B) will no longer be diagonalmatrices.

5. Embodiments and Method for the Disclosed Technology

FIG. 16 is a block diagram representation of a wireless hardwareplatform 1600 which may be used to implement the various methodsdescribed in the present document. The hardware platform 1600 may beincorporated within a base station or a user device. The hardwareplatform 1600 includes a processor 1602, a memory 1604 and a transceivercircuitry 1606. The processor 1602 may execute instructions, e. g., byreading from the memory 1604, and control the operation of thetransceiver circuitry 1606 and the hardware platform 1600 to perform themethods described herein. In some embodiments, the memory, processor andtransceiver may be included in a single chip solution.

FIG. 17 shows an example of a wireless communication system 1700 inwhich a transmitter device 1702 transmits signals to a receiver 1704.The signals may undergo various wireless channels and multipaths, asdepicted. Some reflectors such as buildings and trees may be static,while others such as cars, may be moving scatterers. The transmitterdevice 1702 may be, for example, a mobile phone, a tablet, a computer,or another Internet of Things (IoT) device such as a smartwatch, acamera, and so on. The receiver device 1704 may be a network device suchas the base station. The signals transmitted from the base station tothe transmitter 1702 may experience similar channel degradationsproduced by static or moving scatterers. The techniques described in thepresent document may be implemented by the devices in the wirelesscommunication system 1700.

FIG. 18 is a flowchart for an example method 1800 for wirelesscommunication. The method 1800 may be implemented by a base station in awireless network. The method 1800 includes receiving (1802), by a firstwireless device (e.g., the base station) during a training phase,reference tones using a first number of resource elements from atransmitter of a second wireless device, wherein the first wirelessdevice comprises multiple receiving antennas, estimating (1804), by thefirst wireless device, from the receiving the reference tones, a secondorder statistics of wireless channels between the multiple receivingantennas and the transmitter of the second wireless device, andperforming (1806) channel estimation, during an operational phasesubsequent to the training phase, using the second order statistics andreference tones received on a second number of resource elements,wherein the second number is less than the first number.

As previously described, the training phase may be based on apre-defined number of repetitions of the reference signal transmission,or may be dependent on passage of a time, such as 10 to 20 seconds,during which the training is performed. The estimation operation 1804may be performed by solving for the various equations described herein.

Upon the base station deciding that the training phase is over, then thebase station may begin operating in the operational phase. Thetransition from the training phase to the operational phase may becommunicated to user devices via an over-the-air message transmission.Alternative, the user devices and the base station may keep track ofnumber of reference signal transmissions performed, and enteroperational phase after a pre-defined threshold is exceeded.

As described herein, during operational phase, a reduced number ofresource elements may be used for reference signal transmissions. Insome embodiments, this number may be selected to be a fixed number thatis known to both the transmitter and the receiver. In general, thisnumber is less (or far less) than the first number of reference tonesused during the training phase. As previously discussed, preferablysufficient reference tones are transmitted and received so that theessential parameters of the channel can be extracted.

The second order statistics may be measured in one of two differenttechniques as described here. In one technique, called the averagingmethod, the channel matrix is formulated as an average of autocovarianceof an estimated channel response over all the receiving antennas of thefirst wireless device, at different time instances. In anothertechnique, called, the direct method, includes estimating covariance ofa channel matrix using a direct method in which the channel matrix isformulated to comprise a number of columns, wherein each columncomprises an estimated channel response over all the receiving antennasof the first wireless device at a time instance. Further details aredescribed with reference to equations (1) to (3).

During the operational phase, the method 700 may include performingchannel estimation using an interpolation filter that interpolates froma decimated version of the channel estimate onto the entire channelestimate. Some example embodiments are provided with respect toEquations (5) to (9).

The method 1800 may further include estimating an error covariancematrix representative of an error in the estimated channel response overall of the receiving antennas computed by interpolating channelestimates; and selectively revising, for future use, the second numberof resource elements of reference tones using a measure of the errorcovariance matrix. The measure of error covariance may be the comparisonbetween a sum of squares of diagonal entries with a threshold, asdescribed with respect to Equation (10) and (11).

FIG. 19 is a flowchart for an example method 1900 of wirelesscommunication. The method 1900 may be implemented by a user device (userequipment) in a wireless system. The method includes transmitting(1902), to multiple receive antennas of a first wireless device from atransmit antenna of a second wireless device, during a training phase,reference tones using a first number of resource elements of a wirelesschannel between the transmit antenna and the multiple receive antennas,receiving (1904), at an end of the training phase, an instruction totransmit reference tones using a second number of resource elements, andtransmitting (1906), during an operational phase after the trainingphase, reference tones to the multiple receive antennas of the firstwireless device using the second number of resource elements, whereinthe second number is different from the first number and wherein thesecond number is based on an estimated second order statistics of thewireless channel.

In some embodiments, the method 1900 further includes receiving, duringthe operational phase, another instruction to transmit reference tonesusing a third number of resource elements, wherein the third number isdifferent from the first number and the second number, and transmitting,after receiving the another instruction, reference tones to the firstwireless device using the third number of resource elements. Forexample, the base station may decide that the reduced number ofreference tones may have to be adjusted upwards (if seeing too manyerrors, see equations (9) and (10)) or adjusted lower when the number ofreflectors in the channel is less than what was observed in the trainingphase. As previously described, the lower number of reference signaloverhead can improve the efficiency of a wireless channel by makinggreater bandwidth for data traffic.

FIG. 20 discloses another method 2000 of wireless communication. Themethod 2000 includes estimating (2002), during a training phase, asecond order statistics for a first wireless channel and a secondwireless channel between a transmitter and a receiver comprisingmultiple antennas, wherein the second order statistics is estimatedusing reference tones transmitted on a first number of resourceelements, predicting (2004), during an operational phase subsequent tothe training phase, an estimate of the second wireless channel based onthe second order statistics and an estimate of the first wirelesschannel calculated using reference tones transmitted on a second numberof resource elements, where the second number is less than the firstnumber, and communicating (2006), during the operational phase, over thesecond wireless channel using the estimate of the second wirelesschannel resulting from the predicting, wherein the first wirelesschannel and the second wireless channel include non-overlappingfrequencies.

In some embodiments, the second order statistics may include across-covariance between an estimate of the first wireless channel andan estimate of the second wireless channel. The estimates of the firstand second wireless channels may be obtained during different timeperiods (e.g., training phases that are offset in time with respect toeach other). Sections 2 and 3 have provided additional details used bythe method 2000.

In the methods 1800, 1900 and 2000, the resource elements may representtime slots or subcarriers. The reference tone transmission during thetraining phase may occur on non-contiguous frequencies.

It will be appreciated that technique for reducing the amount oftransmission resources used for reference signal transmissions aredisclosed. Using the disclosed techniques, e.g., determination of secondorder statistics, calculation of an interpolation filter, andinterpolating based on the interpolation filter, can be used to reducethe number of pilot transmissions after an initial training phase inwhich an estimate of the channel and its second order statistics areobtained. The reduced number of transmissions is sufficient as long asthese transmissions provide sufficient information via diversity inangle of arrival, number of transmissions and resources used fortransmissions such that the receiver is able to discern the wirelesschannel characteristics, e.g., dominant reflectors in the channel. Thisreduced set of number may be fixed a priori (e.g., based on a generalknowledge of the expected channel under which the wireless communicationis to operate) or may be changed from time to time based on thedisclosed error measurements. As an illustrative example, if a wirelesschannel is characterized by 8 dominant reflectors, it may be sufficientto send a single reference tone to 8 receive antennas of a base stationfrom a transmit antenna (e.g., a user device) to estimate and predictthe entire channel both in the uplink and in the downlink directions,and at future transmission times.

FIG. 21 is a flowchart for an example method 600 of wirelesscommunication. At 2102, a first communication device receives a numberof subcarriers from a second communication device. Each subcarrierincludes a corresponding reference signal. For example, the firstcommunication device may be Terminal B in FIG. 3B. In some embodiments,the first communication device may be a user equipment (UE) or awireless terminal that is incorporated into a phone, camera, laptop,tablet, Internet of Things (IoT) device or another device capable ofreceived and transmitting wireless signals. In some embodiments, thesecond communication device may be the above-discussed Terminal A. Insome embodiments, the second communication device may be a base station,an access point, Hub 102 (in FIG. 1), macro tower 202 (in FIG. 2), oranother network-side device that is providing communication connectivityto multiple devices such as the first communication device.

In some embodiments, the receiving operation 2102 may occur in a singletime slot of a TDD communication system. Alternatively, the referencesignal transmissions over the subcarriers may be received over multipletime slots. Similarly, in some implementations, multiple referencesignal transmissions on a same subcarrier may be received over multipletimeslots and the results may be averaged to reduce effect of noise onthe various calculations described herein.

At 2102, the first communication device may calculate an inversionfactor for each subcarrier based on the received value of the referencesignal received on respective subcarrier. As described in the presentdocument, e.g., equation (28), a zero forcing technique to ensurenumerical stability of the calculation may be used. The regularized zeroforcing technique, for example, eliminates singularity caused by zero orlow valued denominator of Equation (28).

At 2106, the first communication device may transmit pilot signals backto the second communication device. The pilot signals are scaled by theinversion factor calculated for the subcarrier on which the pilot isbeing transmitted.

As described in the present document, one of the advantageous aspect ofthe method 2100 is that each inversion factor is a scalar. Inversionfactors may be rational numbers or, in general, complex numbersrepresenting both a scale and a phase shift for the subcarrier.

In some embodiments, the pilot signal and the reference signals may bothbe unit signals. For example, the pilot and reference signals may simplymultiply a pre-determined subcarrier level by a factor of 1.Alternatively, other scaling factors may be used. Similarly, thereference signal may also simply multiply a nominal subcarrier level bya scale factor of 1. Other scale factors may be used for multiplying anominal subcarrier value both to the first communication device, andfrom the subcarrier device. These scale factors may, for example, beselected such that the product of “to” and “from” scale factors fornominal subcarrier signal level scaling is 1.

As previously described, averaging may be performed by the firstcommunication device, by repeating inversion factor calculations forsubcarriers over multiple receptions.

FIG. 22 is a flowchart of an example method 2200 of wirelesscommunication. The method 2200 may transmit reference signals andprecoded signals to a first communication device from a secondcommunication device. The first and the second communication devices maybe embodied, for example, as described with respect to method 2100.

At 2202, the second communication device transmits reference signalsusing a number of subcarriers. The reference signals may simply scale byunity a nominal signal level of the subcarrier. Other scale factors maybe used, as described with respect to method 2100. The reference signaltransmissions may be performed in a single time slot. Alternatively,multiple reference signal transmissions may be performed over multipletime slots. The reference signal transmissions may be repeated as oftenas needed, e.g., based on the effectiveness of the subsequent pre-codingto achieve or maintain low error communication.

At 2204, the second communication device receives inversion factors inthe form or scaled pilot signals from the first communication device,where the subcarriers that were used for reference signal transmissionsare scaled by the corresponding inversion factor calculated by the firstcommunication device.

At 2206, the second communication device may estimate the communicationchannel from the second communication device to the first communicationdevice by generating values on all the subcarriers based on theinversion factors that were received on the subset of the subcarriers.For example, simple linear interpolation may be used for reducedcomputational complexity.

After an estimate of the communication channel based on the inversionfactors is obtained, the second communication device may then computer afull channel response that also includes channel response of thereflectors. In some embodiments, second order statistics of the effectof the reflectors may be used for obtaining the full channel response.

In some embodiments, the first communication device is Terminal Bdescribed in the present document and depicted in FIG. 3B. Typically,multiple Terminal Bs may be present in a wireless communication system,with a single Terminal A, and possibly a secondary base station, willprovide wireless access to multiple Terminal B's. In some embodiments,the second communication device is Terminal A described in the presentdocument and depicted in FIG. 3B. With respect to the methods 600 and700, the sparse set of subcarriers over which the reference signals (andsubsequently pilot signals) may be transmitted may include every Mthsubcarrier, such as every 8th or every 16th subcarrier. In someembodiments, a non-uniform group of subcarriers may be used.

In some embodiments, a wireless communication device (e.g., 1600) mayinclude a processor and a transceiver circuitry such that the processorcontrols the operation of inversion factor calculations and thetransceiver circuitry is used to transmit or receive the referencesignals and the pilot signals.

It will be appreciated by one of skill in the art that the describedreceiver-side inversion is a novel approach for computing reciprocitycalibration factors that has low feedback overhead and more favorabledynamic range requirements compared to existing solutions. For a giventone in the frequency domain, only one symbol transmission in eachdirection is required to obtain the reciprocity calibration factor,which greatly reduces overhead compared to feeding back channels as datasamples. As such, it is well suited to enabling efficient bidirectionaltransmission in TDD wireless systems operating with a large number ofboth BS antennas and CPEs.

The following listing of examples provide embodiments that can addressedthe technical problems described in the present document, among otherproblems.

1. A wireless communication method, comprising: receiving, by a firstwireless device during a training phase, reference tones using a firstnumber of resource elements from a transmitter of a second wirelessdevice, wherein the first wireless device comprises multiple receivingantennas; estimating, by the first wireless device, from the receivingthe reference tones, a second order statistics of wireless channelsbetween the multiple receiving antennas and the transmitter of thesecond wireless device; and performing channel estimation, during anoperational phase subsequent to the training phase, using the secondorder statistics and reference tones received on a second number ofresource elements, wherein the second number is less than the firstnumber.

2. The method of example 1, wherein the estimating the second orderstatistics includes estimating covariance of a channel matrix using adirect method in which the channel matrix is formulated to comprise anumber of columns, wherein each column comprises an estimated channelresponse over all the receiving antennas of the first wireless device ata time instance.

3. The method of example 1, wherein the estimating the second orderstatistics includes estimating covariance of a channel matrix using anaveraging method in which the channel matrix is formulated as an averageof autocovariance of an estimated channel response over all thereceiving antennas of the first wireless device, at different timeinstances.

4. The method of example 1, wherein the performing the channelestimation includes: performing the channel estimation by interpolatingchannel estimates from the reference tones received on the second numberof resource elements using an interpolation filter.

5. The method of example 1, further comprising: determining, using thesecond order statistics, an interpolation filter.

6. The method of example 5, wherein the determining the interpolationfilter includes using a decimated version of a covariance matrixcalculated during the training phase.

7. The method of example 6, wherein the determining the interpolationfilter is performed by: determining an eigenvector factorization of thecovariance matrix, wherein the eigenvector factorization is representedas: R_(HH)≈V·D·V*, wherein R_(HH) is the covariance matrix, V is aneigenvectors matrix and D is a diagonal matrix of eigenvalues, and * isa conjugate transpose operation; and estimating the interpolation filteras function of the eigenvectors matrix and a decimated version of theeigenvectors matrix.

8. The method of example 7, wherein the function is represented asC=V·(V)⁻¹, where C represents the interpolation filter, V_(M) is thedecimated version of V.

9. The method of example 5, further including: estimating an errorcovariance matrix representative of an error in the estimated channelresponse over all of the receiving antennas computed by interpolatingchannel estimates; and selectively revising, for future use, the secondnumber of resource elements of reference tones using a measure of theerror covariance matrix.

10. The method of example 9, wherein the measure of the error covariancematrix includes a comparison between a sum of squares of diagonalentries of the error covariance matrix and a threshold.

11. A method of wireless communication, comprising: transmitting, tomultiple receive antennas of a first wireless device from a transmitantenna of a second wireless device, during a training phase, referencetones using a first number of resource elements of a wireless channelbetween the transmit antenna and the multiple receive antennas;receiving, at an end of the training phase, an instruction to transmitreference tones using a second number of resource elements; andtransmitting, during an operational phase after the training phase,reference tones to the multiple receive antennas of the first wirelessdevice using the second number of resource elements, wherein the secondnumber is different from the first number and wherein the second numberis based on an estimated second order statistics of the wirelesschannel.

12. The method of example 11, further including: receiving, during theoperational phase, another instruction to transmit reference tones usinga third number of resource elements, wherein the third number isdifferent from the first number and the second number; and transmitting,after receiving the another instruction, reference tones to the firstwireless device using the third number of resource elements.

13. The method of any of examples 1 to 12, wherein the resource elementscomprise subcarriers.

14. The method of any of examples 1 to 12, wherein the resource elementscomprise time slots.

15. The method of any of examples 1 to 14, wherein the second number ofresource elements is equal to 1.

16. A wireless communication method, implementable by a wirelesscommunication apparatus, comprising: estimating, during a trainingphase, a second order statistics for a first wireless channel and asecond wireless channel between a transmitter and a receiver comprisingmultiple antennas, wherein the second order statistics is estimatedusing reference tones transmitted on a first number of resourceelements; predicting, during an operational phase subsequent to thetraining phase, an estimate of the second wireless channel based on thesecond order statistics and an estimate of the first wireless channelcalculated using reference tones transmitted on a second number ofresource elements, where the second number is less than the firstnumber; and communicating, during the operational phase, over the secondwireless channel using the estimate of the second wireless channelresulting from the predicting; wherein the first wireless channel andthe second wireless channel include non-overlapping frequencies.

17. The method of example 16, wherein the second order statisticscomprises a cross-covariance between an estimate of the first wirelesschannel and an estimate of the second wireless channel.

18. The method of example 17, wherein the estimate of the first wirelesschannel and the estimate of the second wireless channel are obtained atdifferent times.

19. The method of example 16 or 17, wherein the method is implemented bya base station in a frequency division duplexing (FDD) network.

20. The method of example 16 or 17, wherein the method is implemented bya base station in a time division duplexing (TDD) network.

21. The method of any of examples 1 to 20, wherein the reference tonesduring the training phase are transmitted over a non-contiguousspectrum.

22. A method of wireless communication, comprising: receiving, by afirst communication device, a number of subcarriers from a secondcommunication device, each including a corresponding reference signal;calculating an inversion factor for each subcarrier based on a receivedvalue of the corresponding reference signal; and transmitting by thefirst communication device to the second communication device, at leastsome of the subcarriers by scaling a pilot signal using a correspondinginversion factor.

23. The method of example 22, wherein the subcarriers are received in asingle time slot.

24. The method of example 22, wherein the subcarriers are received overmultiple time slots.

25. The method of any of examples 22 to 24, wherein the calculating theinversion factor includes using a regularized zero forcing technique,thereby avoiding singularities in calculations.

26. The method of example 22, wherein the inversion factor is a complexnumber.

27. The method of any of examples 22 to 26, wherein the scaling thepilot signal includes multiplying the pilot signal by the inversionfactor.

28. The method of example 22, wherein the first communication device isa user terminal in a wireless network and the second communicationdevice is a network device in the wireless network.

29. The method of example 25, wherein the calculating the inversionfactor includes evaluating:

${{\overset{\sim}{H}}_{AB}^{- 1} = \frac{H_{AB}^{*}}{{H_{AB}^{*} \cdot H_{AB}} + N_{0}}},$

wherein {tilde over (H)}_(AB) ⁻¹ represents the inversion factor,H*_(AB) represents complex conjugate of the received value, and Norepresents noise variance in received reference signals.

30. The method of example 29, wherein the calculating is repeated andaveraged over multiple received reference signal transmissions for eachsubcarrier.

31. The method of any of examples 22 to 29, wherein the reference signaland the pilot signal are inverse functions of each other.

32. A method of wireless communication, comprising: transmitting, to afirst communication device, from a second communication device, a numberof subcarriers, each subcarrier including a corresponding referencesignal; receiving, from the first communication device, at least some ofthe subcarriers carrying pilot signals scaled by inversions factors forthe subcarriers; and estimating a communication channel between thesecond communication device to the first communication device using theinversion factors.

33. The method of example 32, wherein the estimating the communicationchannel includes interpolating inversion factors at intermediatesubcarriers for which no inversion factors were received from the firstcommunication device.

34. The method of example 32, wherein the corresponding referencesignals transmitted on each subcarriers are identical.

35. The method of example 32, further including: performing a subsequenttransmission from the second communication device to the firstcommunication device by pre-coding using a result of the estimating thecommunication channel.

36. The method of example 35, wherein the pre-coding includesTomlinson-Harashima precoding.

37. The method of example 32, wherein the subcarriers on which referencesignals are transmitted include every Mth subcarrier of thecommunication channel, where M is an integer greater than 1.

38. The method of example 32, wherein the estimating the communicationchannel further includes estimating contributions of reflectors to thecommunication channel using second order statistics.

39. The method of example 32, wherein the inversion factors are complexscalar numbers.

40. A wireless communication apparatus comprising: a processor; anon-transitory memory; and a wireless transceiver, wherein a methodrecited in one or more of examples 1 to 39 is stored in thenon-transitory memory, and wherein the processor is configured toperform the method using the wireless transceiver for transmitting orreceiving signals.

41. The wireless communication apparatus of example 22, wherein thewireless communication apparatus is a base station of a multi-usermulti-input multi-output (MU-MIMO) wireless system.

The disclosed and other embodiments, modules and the functionaloperations described in this document can be implemented in digitalelectronic circuitry, or in computer software, firmware, or hardware,including the structures disclosed in this document and their structuralequivalents, or in combinations of one or more of them. The disclosedand other embodiments can be implemented as one or more computer programproducts, i.e., one or more modules of computer program instructionsencoded on a computer readable medium for execution by, or to controlthe operation of, data processing apparatus. The computer readablemedium can be a machine-readable storage device, a machine-readablestorage substrate, a memory device, a composition of matter effecting amachine-readable propagated signal, or a combination of one or morethem. The term “data processing apparatus” encompasses all apparatus,devices, and machines for processing data, including by way of example aprogrammable processor, a computer, or multiple processors or computers.The apparatus can include, in addition to hardware, code that creates anexecution environment for the computer program in question, e.g., codethat constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, or a combination of one or moreof them. A propagated signal is an artificially generated signal, e.g.,a machine-generated electrical, optical, or electromagnetic signal, thatis generated to encode information for transmission to suitable receiverapparatus.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, and it can bedeployed in any form, including as a standalone program or as a module,component, subroutine, or other unit suitable for use in a computingenvironment. A computer program does not necessarily correspond to afile in a file system. A program can be stored in a portion of a filethat holds other programs or data (e.g., one or more scripts stored in amarkup language document), in a single file dedicated to the program inquestion, or in multiple coordinated files (e.g., files that store oneor more modules, sub programs, or portions of code). A computer programcan be deployed to be executed on one computer or on multiple computersthat are located at one site or distributed across multiple sites andinterconnected by a communication network.

The processes and logic flows described in this document can beperformed by one or more programmable processors executing one or morecomputer programs to perform functions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application specific integrated circuit).

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andany one or more processors of any kind of digital computer. Generally, aprocessor will receive instructions and data from a read-only memory ora random access memory or both. The essential elements of a computer area processor for performing instructions and one or more memory devicesfor storing instructions and data. Generally, a computer will alsoinclude, or be operatively coupled to receive data from or transfer datato, or both, one or more mass storage devices for storing data, e.g.,magnetic, magneto optical disks, or optical disks. However, a computerneed not have such devices. Computer readable media suitable for storingcomputer program instructions and data include all forms of non-volatilememory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices; magnetic disks, e.g., internal hard disks or removable disks;magneto optical disks; and CD ROM and DVD-ROM disks. The processor andthe memory can be supplemented by, or incorporated in, special purposelogic circuitry.

While this patent document contains many specifics, these should not beconstrued as limitations on the scope of an invention that is claimed orof what may be claimed, but rather as descriptions of features specificto particular embodiments. Certain features that are described in thisdocument in the context of separate embodiments can also be implementedin combination in a single embodiment. Conversely, various features thatare described in the context of a single embodiment can also beimplemented in multiple embodiments separately or in any suitablesub-combination. Moreover, although features may be described above asacting in certain combinations and even initially claimed as such, oneor more features from a claimed combination can in some cases be excisedfrom the combination, and the claimed combination may be directed to asub-combination or a variation of a sub-combination. Similarly, whileoperations are depicted in the drawings in a particular order, thisshould not be understood as requiring that such operations be performedin the particular order shown or in sequential order, or that allillustrated operations be performed, to achieve desirable results.

Only a few examples and implementations are disclosed. Variations,modifications, and enhancements to the described examples andimplementations and other implementations can be made based on what isdisclosed.

What is claimed is:
 1. A wireless communication method, implementable bya wireless communication apparatus, comprising: estimating, during atraining phase, a second order statistics for a first wireless channeland a second wireless channel between a transmitter and a receivercomprising multiple antennas, wherein the second order statistics isestimated using reference tones transmitted on a first number ofresource elements; predicting, during an operational phase subsequent tothe training phase, an estimate of the second wireless channel based onthe second order statistics and an estimate of the first wirelesschannel calculated using reference tones transmitted on a second numberof resource elements, where the second number is less than the firstnumber; and communicating, during the operational phase, over the secondwireless channel using the estimate of the second wireless channelresulting from the predicting; wherein the first wireless channel andthe second wireless channel include non-overlapping frequencies.
 2. Themethod of claim 1, wherein the second order statistics comprises across-covariance between an estimate of the first wireless channel andan estimate of the second wireless channel.
 3. The method of claim 2,wherein the estimate of the first wireless channel and the estimate ofthe second wireless channel are obtained at different times.
 4. Themethod of claim 1, wherein the predicting the estimate of the secondwireless channel comprises: computing an average of the second-orderstatistics, wherein the estimate of the second wireless channel is basedon an inverse of the average of the second-order statistics.
 5. Themethod of claim 4, wherein the inverse of the average of thesecond-order statistics is approximated using principal componentanalysis (PCA).
 6. The method of claim 1, wherein the method isimplemented by a base station in a frequency division duplexing (FDD)network.
 7. The method of claim 1, wherein the method is implemented bya base station in a time division duplexing (TDD) network.
 8. The methodof claim 1, wherein the reference tones during the training phase aretransmitted over a non-contiguous spectrum.
 9. A wireless communicationapparatus, comprising: a processor; and a transceiver, wherein theprocessor is configured to: estimate, during a training phase, a secondorder statistics for a first wireless channel and a second wirelesschannel between a transmitter and a receiver comprising multipleantennas, wherein the second order statistics is estimated usingreference tones transmitted on a first number of resource elements, andpredict, during an operational phase subsequent to the training phase,an estimate of the second wireless channel based on the second orderstatistics and an estimate of the first wireless channel calculatedusing reference tones transmitted on a second number of resourceelements, where the second number is less than the first number, whereinthe transceiver is configured to: communicate, during the operationalphase, over the second wireless channel using the estimate of the secondwireless channel resulting from the predicting, and wherein the firstwireless channel and the second wireless channel include non-overlappingfrequencies.
 10. The wireless communication apparatus of claim 9,wherein the second order statistics comprises a cross-covariance betweenan estimate of the first wireless channel and an estimate of the secondwireless channel.
 11. The wireless communication apparatus of claim 10,wherein the estimate of the first wireless channel and the estimate ofthe second wireless channel are obtained at different times.
 12. Thewireless communication apparatus of claim 9, wherein the processor isconfigured, as part of predicting the estimate of the second wirelesschannel, to: compute an average of the second-order statistics, whereinthe estimate of the second wireless channel is based on an inverse ofthe average of the second-order statistics.
 13. The wirelesscommunication apparatus of claim 12, wherein the inverse of the averageof the second-order statistics is approximated using principal componentanalysis (PCA).
 14. The wireless communication apparatus of claim 9, themethod is implemented by a base station in a frequency divisionduplexing (FDD) network or a time division duplexing (TDD) network. 15.The wireless communication apparatus of claim 9, wherein the wirelesscommunication apparatus is a base station of a multi-user multi-inputmulti-output (MU-MIMO) wireless system.
 16. A non-transitorycomputer-readable storage medium having instructions stored thereuponfor wireless communication, the instructions when executed by a dataprocessing apparatus, cause the data processing apparatus to performoperations comprising: estimating, during a training phase, a secondorder statistics for a first wireless channel and a second wirelesschannel between a transmitter and a receiver comprising multipleantennas, wherein the second order statistics is estimated usingreference tones transmitted on a first number of resource elements;predicting, during an operational phase subsequent to the trainingphase, an estimate of the second wireless channel based on the secondorder statistics and an estimate of the first wireless channelcalculated using reference tones transmitted on a second number ofresource elements, where the second number is less than the firstnumber; and communicating, during the operational phase, over the secondwireless channel using the estimate of the second wireless channelresulting from the predicting; wherein the first wireless channel andthe second wireless channel include non-overlapping frequencies.
 17. Thenon-transitory computer-readable storage medium of claim 16, wherein thesecond order statistics comprises a cross-covariance between an estimateof the first wireless channel and an estimate of the second wirelesschannel.
 18. The non-transitory computer-readable storage medium ofclaim 17, wherein the estimate of the first wireless channel and theestimate of the second wireless channel are obtained at different times.19. The non-transitory computer-readable storage medium of claim 16,wherein the data processing apparatus is implemented by a base stationin a frequency division duplexing (FDD) network or a time divisionduplexing (TDD) network.
 20. The non-transitory computer-readablestorage medium of claim 16, wherein the reference tones during thetraining phase are transmitted over a non-contiguous spectrum.